Uncertainty quantification (UQ) is a modern inter-disciplinary science that cuts across traditional research groups and combines statistics, numerical analysis and computational applied mathematics.
When we attempt to simulate complex real-world phenomena (eg the spread of infections, the climate, fluid flows, cell growth, the price of stock options) using mathematical and computer models, there is almost always uncertainty in our predictions. Some of this uncertainty stems from the traditional errors that we make when we use numerical methods on computers to approximate solutions to complex models. However, we frequently also encounter model and input uncertainty. A lack of knowledge about the underlying physical processes and scales in the problem of interest means that we must adopt models that reflect our best understanding but which are actually incorrect. Moreover, most models have inputs which may be unknown, unmeasurable or only indirectly observed.
Numerical, modelling and input uncertainties come in two flavours. Some pheonomena are truely random in nature. In that case, the uncertainty is inherent to the problem and cannot be reduced. On the other hand, epistemic uncertainty arises from a basic lack of knowledge about the processes involved which hypothetically could be reduced with more time and resources.
The science of UQ includes not only the development and analysis of numerical methods for solving forward problems with uncertain inputs (propagating uncertainty in model inputs to model outputs) but also for solving inverse problems (where unknown model inputs are to be estimated from possibly noisy observations of model outputs), the study and analysis of uncertainty in the mathematical models themselves, as well as stochastic and multiscale models. The goal in all UQ computations is to provide a statement, often in terms of a probability, about a quantity of interest when there is some form of uncertainty in the solution process.
The quantity and complexity of data available to researchers continues to increase rapidly. Data Science is the diverse set of attempts to deal with this challenge.
We seek to develop the mathematical methods for working with large and / or complex datasets, such as networks, biological and industrial data.
One example is designing algorithms such as Markov chain Monte Carlo (MCMC) methods so that modern computational resources can be efficiently leveraged.
Another is to obtain theoretical understanding of data based on mathematical analysis of models of the process that generated it. This can lead to application-specific improvements in inference and data analysis.
In Manchester, we are currently actively engaged in research projects related to the following UQ topics:
- Numerical Solution of Stochastic PDEs & PDEs with Random Inputs
- Forward propagation of uncertainty
- Statistical and Bayesian inverse problems
- Stochastic modelling
- MCMC methods
- Multiscale methods for stochastic processes
- Applications in geophysics, life sciences and epidemiology / public health
- Model calibration